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Theorem txindis 22785
Description: The topological product of indiscrete spaces is indiscrete. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
txindis ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}

Proof of Theorem txindis
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 neq0 4279 . . . . . . 7 𝑥 = ∅ ↔ ∃𝑦 𝑦𝑥)
2 indistop 22152 . . . . . . . . . . 11 {∅, 𝐴} ∈ Top
3 indistop 22152 . . . . . . . . . . 11 {∅, 𝐵} ∈ Top
4 eltx 22719 . . . . . . . . . . 11 (({∅, 𝐴} ∈ Top ∧ {∅, 𝐵} ∈ Top) → (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ↔ ∀𝑦𝑥𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
52, 3, 4mp2an 689 . . . . . . . . . 10 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ↔ ∀𝑦𝑥𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥))
6 rsp 3131 . . . . . . . . . 10 (∀𝑦𝑥𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → (𝑦𝑥 → ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
75, 6sylbi 216 . . . . . . . . 9 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑦𝑥 → ∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)))
8 elssuni 4871 . . . . . . . . . . . . . 14 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ({∅, 𝐴} ×t {∅, 𝐵}))
9 indisuni 22153 . . . . . . . . . . . . . . 15 ( I ‘𝐴) = {∅, 𝐴}
10 indisuni 22153 . . . . . . . . . . . . . . 15 ( I ‘𝐵) = {∅, 𝐵}
112, 3, 9, 10txunii 22744 . . . . . . . . . . . . . 14 (( I ‘𝐴) × ( I ‘𝐵)) = ({∅, 𝐴} ×t {∅, 𝐵})
128, 11sseqtrrdi 3972 . . . . . . . . . . . . 13 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ⊆ (( I ‘𝐴) × ( I ‘𝐵)))
1312ad2antrr 723 . . . . . . . . . . . 12 (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑥 ⊆ (( I ‘𝐴) × ( I ‘𝐵)))
14 ne0i 4268 . . . . . . . . . . . . . . . . . . . 20 (𝑦 ∈ (𝑧 × 𝑤) → (𝑧 × 𝑤) ≠ ∅)
1514ad2antrl 725 . . . . . . . . . . . . . . . . . . 19 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) ≠ ∅)
16 xpnz 6062 . . . . . . . . . . . . . . . . . . 19 ((𝑧 ≠ ∅ ∧ 𝑤 ≠ ∅) ↔ (𝑧 × 𝑤) ≠ ∅)
1715, 16sylibr 233 . . . . . . . . . . . . . . . . . 18 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 ≠ ∅ ∧ 𝑤 ≠ ∅))
1817simpld 495 . . . . . . . . . . . . . . . . 17 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ≠ ∅)
1918neneqd 2948 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → ¬ 𝑧 = ∅)
20 simpll 764 . . . . . . . . . . . . . . . . . . 19 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ∈ {∅, 𝐴})
21 indislem 22150 . . . . . . . . . . . . . . . . . . 19 {∅, ( I ‘𝐴)} = {∅, 𝐴}
2220, 21eleqtrrdi 2850 . . . . . . . . . . . . . . . . . 18 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 ∈ {∅, ( I ‘𝐴)})
23 elpri 4583 . . . . . . . . . . . . . . . . . 18 (𝑧 ∈ {∅, ( I ‘𝐴)} → (𝑧 = ∅ ∨ 𝑧 = ( I ‘𝐴)))
2422, 23syl 17 . . . . . . . . . . . . . . . . 17 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 = ∅ ∨ 𝑧 = ( I ‘𝐴)))
2524ord 861 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (¬ 𝑧 = ∅ → 𝑧 = ( I ‘𝐴)))
2619, 25mpd 15 . . . . . . . . . . . . . . 15 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑧 = ( I ‘𝐴))
2717simprd 496 . . . . . . . . . . . . . . . . 17 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ≠ ∅)
2827neneqd 2948 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → ¬ 𝑤 = ∅)
29 simplr 766 . . . . . . . . . . . . . . . . . . 19 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ∈ {∅, 𝐵})
30 indislem 22150 . . . . . . . . . . . . . . . . . . 19 {∅, ( I ‘𝐵)} = {∅, 𝐵}
3129, 30eleqtrrdi 2850 . . . . . . . . . . . . . . . . . 18 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 ∈ {∅, ( I ‘𝐵)})
32 elpri 4583 . . . . . . . . . . . . . . . . . 18 (𝑤 ∈ {∅, ( I ‘𝐵)} → (𝑤 = ∅ ∨ 𝑤 = ( I ‘𝐵)))
3331, 32syl 17 . . . . . . . . . . . . . . . . 17 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑤 = ∅ ∨ 𝑤 = ( I ‘𝐵)))
3433ord 861 . . . . . . . . . . . . . . . 16 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (¬ 𝑤 = ∅ → 𝑤 = ( I ‘𝐵)))
3528, 34mpd 15 . . . . . . . . . . . . . . 15 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑤 = ( I ‘𝐵))
3626, 35xpeq12d 5620 . . . . . . . . . . . . . 14 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) = (( I ‘𝐴) × ( I ‘𝐵)))
37 simprr 770 . . . . . . . . . . . . . 14 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (𝑧 × 𝑤) ⊆ 𝑥)
3836, 37eqsstrrd 3960 . . . . . . . . . . . . 13 (((𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵}) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (( I ‘𝐴) × ( I ‘𝐵)) ⊆ 𝑥)
3938adantll 711 . . . . . . . . . . . 12 (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → (( I ‘𝐴) × ( I ‘𝐵)) ⊆ 𝑥)
4013, 39eqssd 3938 . . . . . . . . . . 11 (((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) ∧ (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥)) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵)))
4140ex 413 . . . . . . . . . 10 ((𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ (𝑧 ∈ {∅, 𝐴} ∧ 𝑤 ∈ {∅, 𝐵})) → ((𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
4241rexlimdvva 3223 . . . . . . . . 9 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (∃𝑧 ∈ {∅, 𝐴}∃𝑤 ∈ {∅, 𝐵} (𝑦 ∈ (𝑧 × 𝑤) ∧ (𝑧 × 𝑤) ⊆ 𝑥) → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
437, 42syld 47 . . . . . . . 8 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑦𝑥𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
4443exlimdv 1936 . . . . . . 7 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (∃𝑦 𝑦𝑥𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
451, 44syl5bi 241 . . . . . 6 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (¬ 𝑥 = ∅ → 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
4645orrd 860 . . . . 5 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → (𝑥 = ∅ ∨ 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
47 vex 3436 . . . . . 6 𝑥 ∈ V
4847elpr 4584 . . . . 5 (𝑥 ∈ {∅, (( I ‘𝐴) × ( I ‘𝐵))} ↔ (𝑥 = ∅ ∨ 𝑥 = (( I ‘𝐴) × ( I ‘𝐵))))
4946, 48sylibr 233 . . . 4 (𝑥 ∈ ({∅, 𝐴} ×t {∅, 𝐵}) → 𝑥 ∈ {∅, (( I ‘𝐴) × ( I ‘𝐵))})
5049ssriv 3925 . . 3 ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ {∅, (( I ‘𝐴) × ( I ‘𝐵))}
519toptopon 22066 . . . . . . 7 ({∅, 𝐴} ∈ Top ↔ {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)))
522, 51mpbi 229 . . . . . 6 {∅, 𝐴} ∈ (TopOn‘( I ‘𝐴))
5310toptopon 22066 . . . . . . 7 ({∅, 𝐵} ∈ Top ↔ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵)))
543, 53mpbi 229 . . . . . 6 {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵))
55 txtopon 22742 . . . . . 6 (({∅, 𝐴} ∈ (TopOn‘( I ‘𝐴)) ∧ {∅, 𝐵} ∈ (TopOn‘( I ‘𝐵))) → ({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵))))
5652, 54, 55mp2an 689 . . . . 5 ({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵)))
57 topgele 22079 . . . . 5 (({∅, 𝐴} ×t {∅, 𝐵}) ∈ (TopOn‘(( I ‘𝐴) × ( I ‘𝐵))) → ({∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ 𝒫 (( I ‘𝐴) × ( I ‘𝐵))))
5856, 57ax-mp 5 . . . 4 ({∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵}) ∧ ({∅, 𝐴} ×t {∅, 𝐵}) ⊆ 𝒫 (( I ‘𝐴) × ( I ‘𝐵)))
5958simpli 484 . . 3 {∅, (( I ‘𝐴) × ( I ‘𝐵))} ⊆ ({∅, 𝐴} ×t {∅, 𝐵})
6050, 59eqssi 3937 . 2 ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (( I ‘𝐴) × ( I ‘𝐵))}
61 txindislem 22784 . . 3 (( I ‘𝐴) × ( I ‘𝐵)) = ( I ‘(𝐴 × 𝐵))
6261preq2i 4673 . 2 {∅, (( I ‘𝐴) × ( I ‘𝐵))} = {∅, ( I ‘(𝐴 × 𝐵))}
63 indislem 22150 . 2 {∅, ( I ‘(𝐴 × 𝐵))} = {∅, (𝐴 × 𝐵)}
6460, 62, 633eqtri 2770 1 ({∅, 𝐴} ×t {∅, 𝐵}) = {∅, (𝐴 × 𝐵)}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844   = wceq 1539  wex 1782  wcel 2106  wne 2943  wral 3064  wrex 3065  wss 3887  c0 4256  𝒫 cpw 4533  {cpr 4563   cuni 4839   I cid 5488   × cxp 5587  cfv 6433  (class class class)co 7275  Topctop 22042  TopOnctopon 22059   ×t ctx 22711
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-topgen 17154  df-top 22043  df-topon 22060  df-bases 22096  df-tx 22713
This theorem is referenced by: (None)
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